Determining Hilbert modular forms by central values of Rankin–Selberg convolutions: the weight aspect

Abstract

The purpose of this paper is to prove that a primitive Hilbert cusp form g is uniquely determined by the central values of the Rankin-Selberg L-functions L(f ⊗ g, 12 ), where f runs through all primitive Hilbert cusp forms of weight k for infinitely many weight vectors k. This result is a generalization of the work of Ganguly, Hoffstein, and Sengupta [3] to the setting of totally real number fields, and it is a weight aspect analogue of the authors own work [6].

Non-vanishing of Rankin-Selberg convolutions for Hilbert modular forms

“…More precisely, let g ∈ Π l (n) for some fixed l ∈ 2N n and n ⊂ O F . By obtaining an asymptotic formula for the harmonic sum of the values L( 1 2 , f ⊗ g) as f varies in Π k (O F ) (which the authors showed in [11,Proposition 2.4]) and using the subconvexity bound L(…”

Automorphe Formen und L-Funktionen

“…To estimate the second moment of Rankin-Selberg L-functions, the Voronoi summation formula is based on the class number one assumption. Thus, for a real number field with arbitrary class number a new version of this has to be proven, a Kloosterman sum for totally real number fields of arbitrary class number is given in [HT16].…”

“…Any f ∈ S k (c) has a Fourier expansion We will consider a subspace of the cusp forms, the so called newforms S k (c). This space is defined as the orthogonal complement with respect to the inner product of the oldforms where the oldforms S k (c) are the cusp forms coming from lower level, see [HT16,page 4].…”

“…Now we are able to define the Rankin-Selberg L-function and state some results. We follow the terminology used in [HT16] and [L79]. Let q = (q) be prime and c a squarefree integral ideal either coprime to q or equal to q.…”

“…The case that c and q are coprime is treated in [HT16]. Thus, we assume c = q = (c), c ∈ F and follow the paper of Li [L79].…”